Subgroup ($H$) information
| Description: | $C_3^4.D_6^2:D_6$ |
| Order: | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,8,16,2,3,7,11,6,4,13,14,15,12,5,17,18,9,10)(19,24)(20,23)(21,22), (3,14,9) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.S_3^5:C_2$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^6.C_6.C_2.C_6.C_2^6$ |
| $W$ | $C_3^6.(D_4\times D_6)$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^4.D_6^2:D_6$ |
| Normal closure: | $C_3^4.S_3^5:C_2$ |
| Core: | $C_3^4.S_3^3$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.S_3^5:C_2$ |